The Gaussian integral is the integral of the function eโx2 over the entire real line, and it is given by:
โซโโโโeโx2dx=ฯโ First, we square the integral:
(โซโโโโeโx2dx)2=โซโโโโโซโโโโeโx2eโy2dxdy Next, we change to polar coordinates:
โซ0โโโซ02ฯโeโr2rdฮธdr=ฯ To evaluate this integral, we use the substitution u=r2 and du=2rdr, which gives:
21โโซ0โโeโudu=21โ Therefore, we have:
(โซโโโโeโx2dx)2=ฯโโซโโโโeโx2dx=ฯโ And this completes the proof.